Start by combining the fractions:
[tex]\dfrac{\cos\alpha}{1+\sin\alpha}\cdot\dfrac{1-\sin\alpha}{1-\sin\alpha}+\dfrac{\cos\alpha}{1-\sin\alpha}\cdot\dfrac{1+\sin\alpha}{1+\sin\alpha}[/tex]
[tex]\dfrac{\cos\alpha(1-\sin\alpha)+\cos\alpha(1+\sin\alpha)}{(1+\sin\alpha)(1-\sin\alpha)}[/tex]
[tex]\dfrac{2\cos\alpha}{1-\sin^2\alpha}[/tex]
Recall the Pythagorean identity:
[tex]\dfrac{2\cos\alpha}{\cos^2\alpha}[/tex]
Then cancel a factor of [tex]\cos\alpha[/tex] and use the definition of secant:
[tex]\dfrac2{\cos\alpha}=\boxed{2\sec\alpha}[/tex]
